Pure square root finding

7

1. The complex number \(u\) is given by \(u = 8 - 15 \mathrm { i }\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(| z - 2 - i | \leqslant 2\) and \(0 \leqslant \arg ( z - i ) \leqslant \frac { 1 } { 4 } \pi\).
   [0pt] [4]
   \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 9 x - 8 } { ( x + 2 ) ( 2 x - 1 ) }\).
   1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x + 2 } + \frac { C } { 2 x - 1 }\).
   2. Hence show that \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 6 + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 7 } \right)\).

10

1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
   [0pt] [5] If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The complex number \(u\) is given by \(u = 10 - 4 \sqrt { 6 } \mathrm { i }\).
Find the two square roots of \(u\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.

3 The square roots of 24-7i can be expressed in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7 \mathrm { i }\) in exact Cartesian form.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-04_2717_35_141_2012} The variables \(x\) and \(y\) satisfy the equation \(k y = \mathrm { e } ^ { c x }\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(2.80,0.372\) ) and ( \(5.10,2.21\) ), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures.

3 The square roots of 6-8i can be expressed in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8 \mathrm { i }\) in exact Cartesian form.
\includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-05_2725_35_99_20}

2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).

4 Use an algebraic method to find the square roots of the complex number 21-20i.

3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

3 Use an algebraic method to find the square roots of \(3 + ( 6 \sqrt { 2 } )\) i. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.

1 In this question you must show detailed reasoning.
Find the square roots of \(24 + 10 \mathrm { i }\), giving your answers in the form \(a + b \mathrm { i }\).

1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).